Problem 77 - Beginning and Intermediate Algebra

Question

Solve. $$x^{1 / 3}=-2$$

Step-by-Step Solution

Answer

The solution to the given equation \(x^{\frac{1}{3}} = -2\) is \(x = -8\).

Step 1: Write given equation

The given equation is \(x^{\frac{1}{3}} = -2\).

Step 2: Raise both sides of the equation to the power of 3

To eliminate the cube root, we will raise both sides of the equation to the power of 3: \[\left( x^{\frac{1}{3}} \right)^3 = (-2)^3\]

Step 3: Simplify the equation

After raising both sides to the power of 3, we simplify the equation: \[x = -8\] So, the solution to the given equation \(x^{1 / 3} = -2\) is \(x = -8\).

Key Concepts

Cube Root

A cube root of a number is another number that, when multiplied by itself three times, results in the original number. This is expressed mathematically as the inverse operation of raising a number to the power of three. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In terms of notation, the cube root of \(x\) is denoted as \(x^{\frac{1}{3}}\).

When dealing with cube roots, it is important to understand that they can produce both positive and negative results. This is because multiplying three negative factors will also result in a negative product. For instance, the cube root of -8 is -2 because \((-2) \times (-2) \times (-2) = -8\). Understanding this concept helps in solving equations like \(x^{\frac{1}{3}} = -2\).

Negative Numbers

Negative numbers represent values less than zero and are denoted with a minus sign in front of them. They frequently appear in real-world contexts such as temperatures below freezing or owing money. In mathematics, negative numbers follow their own set of rules especially when involving operations like addition, subtraction, multiplication, or division.

When multiplying two negative numbers, the result is positive. However, when multiplying an odd number of negative numbers together, as seen in cubing them, the result is again negative. In our solved problem, \(-2\) raised to the power of three remains negative, resulting in \(-8\). This is crucial when working with equations containing cube roots or odd powers, as they will affect the final sign of your solution.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It describes repeated multiplication of the base, defined by the exponent. For example, the expression \(a^3\) denotes \(a \times a \times a\). When performing calculations involving exponentiation, understanding how to apply these operations correctly is key.

In our context, raising a number with a fractional exponent, like \(x^{\frac{1}{3}}\), means finding the cube root of \(x\). Conversely, to eliminate a fractional exponent, we reverse the operation by raising both sides of the equation to the reciprocal exponent. Hence, \(x^{\frac{1}{3}} = -2\) becomes \((x^{\frac{1}{3}})^3 = (-2)^3\), ultimately solving for \(x\).

Simplifying Equations

Simplifying equations involves combining like terms and performing operations to make the equation easier to solve. The goal is to isolate the variable on one side. This can involve a variety of techniques like distributing, factoring, or clearing fractions and exponents.

In this exercise, simplifying the equation required us to raise both sides by the power of 3. This step effectively "cancels out" the cube root, leaving a simple expression that is easy to solve. Thus, \((x^{\frac{1}{3}})^3 = (-2)^3\) simplifies to \(x = -8\).

• Always ensure that all similar terms are combined and operations are carefully executed to avoid mistakes.
• It's important to check back after every operation to ensure equations balance correctly on both sides.